The intense back-and-forth between the justices and the lawyers who made arguments revealed that this question is incredibly complex, involving a lot of case law and, as one lawyer put it, some "loosey-goosey analysis."

But one of the more interesting, somewhat hidden questions is whether a justice can trade her robe for a cassock. As Sonia Sotomayor pointed out, the court has traditionally refrained from trying to figure out what's in someone's heart—if a person claims to have a certain belief, the court takes that claim at face value. But she also pointed out that this makes it nearly impossible to determine whether a company like Conestoga Wood or Hobby Lobby is actually "religious."

"Let's assume a business that sells five percent of religious books, doesn't play Christmas music, doesn't give off ... Sunday, you know, does nothing else religiously," Sotomayor said. "That's the most dangerous piece. That's the one we've resisted in all our exercise jurisprudence: to measure the depth of someone's religious beliefs."

In other words, if a company is just "a little bit religious," could it still claim to have a moral objection to providing birth control coverage or following any other kind of law?

The plaintiff's lawyer, Paul Clement, replied that the court has always had some latitude in figuring this out. "The idea that you don't really second-guess the person's belief, but you can contest sincerity. You know, you have people who are arrested in possession of large quantities of marijuana and they assert that they belong to the church of marijuana, and those cases do get litigated and they get rejected," he said.

So far, two different terms have been used: "depth of belief," which roughly stands for spiritual intensity, and "sincerity," which is a way of judging whether someone is making their beliefs up for legal convenience. In part, the justices' job will come down to disentangling these two concepts on the way to answering one of the biggest legal issues in the case: whether the Affordable Care Act places a "burden" on these companies that prevents them (or their owners) from practicing their religion.

When the government's lawyer, Donald Verrilli, tried to tackle this question, the courtroom briefly turned into a divinity classroom. He argued that since the law doesn't ask companies to hand out birth control themselves, it's not a burden. "The requirement is to purchase insurance which enables actions by others," he said. "You're really closer to the tax situation than to imposing a direct obligation to act."

Justice Alito disagreed. "Isn't that really a question of theology or moral philosophy, which has been debated by many scholars and adherents to many religions?" he asked. "[Person] A does something that [person] B thinks is immoral. How close a connection does there have to be ... in order for B to be required to refrain from doing that action?"

"It's true that it's a difficult question. But it isn't—" Verrilli started.

"It is a religious question and it's a moral question. And you want us to provide a definitive secular answer to it?" Alito shot back.

All of this goes back to the question of burden. It will be the justices' job to figure out what this case is: a question of moral disagreement, no matter how sincere; or a question of burden, meaning that the law prevents Hobby Lobby, Conestoga Wood, and their owners from freely practicing their faith.

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Emma Green is a staff writer at ​The Atlantic, where she covers politics, policy, and religion.